Quality Engineering PRE-CONTROL AND SOME SIMPLE ALTERNATIVES

This article was downloaded by: [Canadian Research Knowledge Network] On: 23 November 2010 Access details: Access Details: [subscription number 918588849] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 3741 Mortimer Street, London W1T 3JH, UK

Quality Engineering

Publication details, including instructions for authors and subscription information:

PRE-CONTROL AND SOME SIMPLE ALTERNATIVES

Stefan H. Steinera a Department of Statistics and Actuarial Sciences, University of Waterloo, Waterloo, Canada

To cite this Article Steiner, Stefan H.(1997) 'PRE-CONTROL AND SOME SIMPLE ALTERNATIVES', Quality Engineering, 10: 1, 65 -- 74 To link to this Article: DOI: 10.1080/08982119708919110 URL:

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

Quality Engineering, 10(1), 65-74 (1997-98)

Downloaded By: [Canadian Research Knowledge Network] At: 18:16 23 November 2010

PRE-CONTROL AND SOME SIMPLE ALTERNATIVES

Stefan H. Steiner

Department of Statistics and Actuarial Sciences University of Waterloo

Waterloo, Canada N2L 3G1

Key Words

Acceptance control charts; Grouped data; Modified h e control; Operating characteristic; Process capability; Stoplight control.

Introduction

Pre-control, sometimes called stoplight control, was developed to monitor the proportion of nonconforming units produced in a manufacturing process. Implementation is typically very straightforward. All test units are classified into one of three groups: green, yellow, or red, where the colors loosely correspond to good, questionable, and poor quality products (see Fig. 1). The number of green, yellow, and red units observed in a small sample determines when to stop and adjust the process. The goal of Pre-control is to detect when the proportion of nonconforming units produced becomes too large. Thus, he-control schemes monitor the process to ensure that process capability (C,,)remains large.

Pre-control was initially proposed in 1954 (see Refs. 1 and 2 for more details) as an easier alternative to Shewhart charts. However, since that time, at least three different versions of Pre-control have been suggested in the literature. In this article the three versions are called

Classical Pre-control * Two-stage Pre-control

Modified Pre-control

Classical Pre-control refers to the original formulation of Pre-control as described by Shainin and Shainin (I) and Traver (3). Two-stage Pre-control is a modification discussed by Salvia (4) that improves the method's operating characteristics by taking an additional sample if the initial sample yields ambiguous results. Modified Pre-control, on the other hand, as suggested by Gurska and Heaphy (5), represents a departure from the philosophy of Classical and Two-stage Pre-control. Modified Pre-control attempts to compromise between the design philosophy of Shewharttype control charts and the simplicity of application of Precontrol.

Pre-control schemes are defined by their group classification procedure, their decision criteria, and their qualification procedure. The qualification procedure specifies the required results of an initial intensive sampling scheme used to determine if Precontrol is appropriate for the given application. For all three versions of he-control, a process passed the qualification if five consecutive green units are observed. As all three versions of Pre-control have the same qualification procedure, it is not discussed in more detail in this article. The three versions of Pre-control differ most substantially in their group classification method. Classical Pre-control and Two-stage Pre-control base the classification of units on engineering tolerance or specification limits. A unit is classified as green if its quality dimension of interest falls into the central half of the tolerance range. A yellow unit has a quality dimension that falls into the remaining tolerance range, and a red unit falls outside the tolerance range. Assuming, without loss of

Copyright 1997 by Marcel Dekker. Inc.

Downloaded By: [Canadian Research Knowledge Network] At: 18:16 23 November 2010

Ral Yellow

-1

-3

LSL

Green

Yellow Red

Measurement J

1

USL

Figure 1. Pre-control classification criteria.

generality, that the upper and lower specification limits (USL and LSL) are 1 and -1, respectively, group classification is'based on the endpoints: r = [-I, -0.5, 0.5, I]. Figure 1 illustrates this classification scheme. The colored circle can be used for the ease of the operators as a dial indicator overlay (I).

Let the quality dimension of interest be Y. Then, given a probability density function for observations fCy), the group probabilities for Classical and Two-stage Pre-control are given by

J Y i fired) = P, =

f(y) dy + f(y) dy. Y-1

Modified Pre-control, on the other hand, classifies units using control limits, as defined for Shewhart charts, rather than tolerance limits. This change indicates a fundamental difference and makes modified Pre-control much more like a Shewhart chart than like Classical he-control. Equations (2) give the group probabilities for the Modified Pre-control procedure. Note that to avoid confusion, throughout this article the current process mean and standard deviation are denoted j~and a,whereas estimates of the in-control mean and standard deviation used to set the control limits for Modified Pre-control and Shewhart-type charts are

denoted p, and a,.

STEINER

The group probabilities given by Eqs. (I) and (2) are the same when p, = 0 and a, = 113. This makes sense because in that case, the control limits and the tolerance limits are the same. It has been suggested that Classical Pre-control and Two-stage Pre-control are only applicable if the current process spread (six process standard deviations) covers less than 88% of the tolerance range (3). With specification limits at f I , as defined previously, this con-

dition corresponds to the constraint a < 0.29333.

The second important difference between the three Precontrol versions is their decision criteria. Classical Precontrol bases the decision to continue operation or to adjust the process on only one or two sample units. The decision rules are given as follows:

Sample two consecutive pans A and B

If part A is green, continue operation (no need to measure B). If part A is yellow, measure part B. If part B is green, continue operation, otherwise stop and adjust process. If part A is red, stop and adjust process (no need to measure B).

The decision procedure for Two-stage Pre-control and Modified Precontrol is more complicated. If the initial two observations do not provide clear evidence regarding the state of the process, additional observations (up to three more) are taken. The decision procedure for Two-stage and Modified Pre-control is given as follows:

Sample two consecutive parts.

If either part is red, stop process and adjust. If both parts are green, continue operation. If either or both of the parts are yellow, continue to sample up to three more units. Continue operation if the combined sample contains three green units, and stop the process if three yellow units or a single red unit are observed.

The advantage of this more complicated decision procedure is that more information regarding the state of the process is obtained and, thus, decision errors are less likely. The disadvantage is that, on average, larger sample sizes are needed to make a decision regarding the state of the process. Table 1 summarizes the comparison of the three versions of he-control. Clearly, the different versions of Pre-control are not the same in purpose and ease of implementation. By design, Classical Precontrol and Twostage Pre-control tolerate some deviation in the process mean, so long as the proportion nonconforming does not

Downloaded By: [Canadian Research Knowledge Network] At: 18:16 23 November 2010

PRE-CONTROL AND SIMPLE ALTERNATIVES

Table 1. Comparison of Pre-control Versions

GROUP

PRE-CONTROL CLASSIFICATION DECISION CRITERIA

VERSION

BASED ON

BASED ON

Classical Two-stage Modified

Tolerance limits Tolerance limits Control limits

Two observations Five observations Five observations

become too.large. In this sense, Classical and Two-stage Pre-control are very similar to acceptance control charts. In addition, Classical and Two-stage Pre-control can be quickly implemented because they do not require estimates of the current process mean and standard deviation to set their grouping criterion. By contrast, the goal of Modified Pre-control is to detect deviations from stability. As a result, mean drifts are not tolerated, and Modified Pre-con-

trol charts are similar to Shewhart-type charts such as X

charts. In addition, Modified Pre-control, like a Shewhart chart, requires estimates of the current process parameters to determine the control limits.

Comparison with Traditional Control Charting Methods

In this section, the three Pre-control versions are compared with the appropriate traditional control charting techniques. In addition, the performance of Pre-control under some special situations is explored. Mackertich (6) and Ermer and Roepke (2) compare Classical and Two-stage

Pre-control with X and R control charts, but this is an

inappropriate comparison because, as explained in the previous section, the charts have a different purpose. Here,

Classical and Two-stage Pre-control are compared with acceptance control charts (ACCs) because both these types of monitoring schemes are designed to signal only if the proportion nonconforming becomes unacceptably high. An ACC is designed to monitor a process when the process variability is much smaller than the specification (tolerance) spread (7). Under that assumption, moderate drifts in the mean (from the target value) are tolerable, as they do not yield a significant increase in the proportion of nonconforming units. Like Classical and Two-stage Pre-control, ACCs are based on engineering specification limits. However, ACC limits are derived based on a distributional assumption and require a known and constant process standard deviation. ACC limits are usually set assuming a normal process, although, if justified, other assumptions could be made. Modified Pre-control, on the other hand, is compared with Shewhart charts, because both charts are designed to monitor the process for stability.

All Pre-control schemes differ from traditional control charting techniques in a number of ways. The first obvious difference is that Pre-control uses only information from the classified (or grouped) observations, whereas uaditional control charts ( X charts and ACCs) use variables data. Grouping the data results in decision rules that are easy to implement, but, clearly, the grouping also discards some information. This loss of information can be quantified by calculating the expected statistical (Fisher) information available in the Pre-control grouped observations. Kulldorff (8) and Steiner et al. (9) discuss calculating the Fisher information available in grouped data. Figure 2 shows the expected information about the mean and standard deviation for various parameter values using the Classical Pre-control classification criterion r = [-I. -0.5, 0.5, 11. The plots are scaled so that variables data (infinite

Figure 2. Expected information about p (left, assume a = 0.2933) and o (right, assume p = 0). t = [-I, -0.5, 0.5, 11.

Downloaded By: [Canadian Research Knowledge Network] At: 18:16 23 November 2010

number of groups) would provide an expected information content of unity for all values of p and o. The expected information about the p graph is symmetric about p = 0; thus, negative mean values are not shown. Figure 2 shows that the group classification scheme used in Pre-control is not very efficient when p is close to zero andlor o is small. The small amount of information available when p equals zero is not a major concern however if, at that mean value, it is very easy to conclude that the process should continue operation. This would be the case, for example, if the process standard deviation is small compared with the specification range. The next section explores this issue further.

Pre-control and traditional control charts also differ in their decision procedures. The effectiveness of the decision procedures can be compared through their operating characteristic (OC) curves. An OC curve shows the probability a process monitoring scheme or control chart signals (or fails to signal) for different parameter values. Ryan (7)

derives OC curves for ACCs and X charts. The probability

that Classical and Two-stage Pre-control schemes continue operating, denoted Paw,, can be found by calculating the probability of each combination of green and yellow units that leads to acceptance. Salvia (4) gives the probability of acceptance for Classical and Two-stage Pre-control as

+ P,,,,, (Classical) = Pg PyPg,

(3)

+ Pa,, (Two-stage) = (Pg Py)2- ZPgPy(1 - Pg3 - 3Pg2Py)

- Py2 (1 - Pg3),

(4)

where Pg = P(green) and Py = P(yellow) are given by Eqs. (1).

Figure 3 compares the OC curves for Classical Precontrol and Two-stage Pre-control, as derived from Eqs. (3) and (4), and an ACC with sample size n = 2. Figure 3 shows that an ACC with sample size n = 2 is superior to Classical Pre-control because it has significantly better power to detect mean shifts. Similarly, the Two-stage Precontrol scheme is superior to an ACC with n = 2. Note that different o values simply shift the OC curves horizontally without changing the ranking. Figure 3 assumes a process whose quality characteristic is approximately normally distributed. A different underlying distribution for the quality characteristic may dramatically change the probabilities of acceptance, but, generally, all the considered procedures are affected similarly.

Two-stage Pre-control has the best operating characteristics and is, thus, preferred over Classical Pre-control. However, in most situations, Two-stage Pre-control re-

STEINER

M a n Shift in SigmaUniu

Figure 3. OC curves of Classical and Two-stage Pre-control and ACC with n = 2, N(p, o = 0.2933) process.

quires sample sizes larger than two to make a decision. As given previously, Two-stage Pre-control uses a partially sequential decision procedure and requires sample sizes of between one and five units. In the Appendix, the average sampling number of the Two-stage Pre-control procedure is derived and denoted by E(n). E(n) depends on the group probabilities, given by Eq. (I), that are, in turn, dependent on the process parameters p and o.Figure 4 shows plots of E(n) versus the process mean when the process is assumed to have a Normal distribution.

These results suggest a comparison of Two-stage Precontrol with ACCs having larger sample sizes. Figure5 gives OC curves of Two-stage Pre-control and ACCs with sample sizes of three, four, and five. Clearly, of the compared curves, the Pre-control procedure has the smallest power to detect mean shifts. In addition, ACCs are likely easier to implement because they require fixed sample sizes. On the other hand, the power of Two-stage Precontrol is fairly competitive with an ACC with n = 3, and ACCs have the disadvantage of requiring an estimate of u, an assumption that o does not change, and precise or variables measurements rather than grouped data. In addition, Two-stage Pre-control requires a smaller sample size, on average, when the process is centered at p = 0.In conclusion, Two-stage he-control appears a reasonable alternative to ACCs when little process knowledge is available or if estimating the process mean and standard deviation is expensive.

PRE-CONTROL AND SIMPLE ALTERNATIVES

Downloaded By: [Canadian Research Knowledge Network] At: 18:16 23 November 2010

Figure 4. Average sample size for Two-stage Pre-control as a function of the process mean a = 0.29333 on left, a = 0.2 on right, N(p, a) process.

Modified Pre-control, unlike Classical and Two-stage

Pre-control, is designed to monitor the stability of a pro-

cess and, thus, has the same philosophy as an X control chart. Modified Pre-control and X charts ideally detect any

drift in the process mean. The operating characteristics of

Modified Pre-control can be determined using Eqs. (4) and

(2) with p = 0 and o = 113. Table 2 shows a compari-

son of Modified Pre-control and traditional X control

charts using various sample sizes. The average sample size

of Modified Pre-control can be determined through Eq. (Al) by calculating E(n; p = 0,o = 113). A plot of the

-- IL 0.6

< .5- 0.5

I - 0.4 - 0.3

ACC n d

ACC n 4

M a Shin hSigma Units

Figure 5. Comparison of Two-stage Pre-control and ACC with n = 3, n = 4, n = 5, N(p, a = 0.29333) process.

average sample size for Modified Pre-control would be similar to Figure 4.

Table 2 shows that Modified Pre-control has a very high false alarm rate that is an order of magnitude larger than

that for % charts. When the process is stable (p = 0), a

Modified Pre-control chart signals a problem on average almost 2.4% of the time. This false alarm rate is too high, as searching for assignable causes is usually time-consuming and expensive, and too many false alarms reduces the credibility of the control chart greatly and may lead to it being ignored.

We now turn to an examination of when Pre-control is applicable. A drawback of Classical and Two-stage Precontrol schemes is that they are designed to prevent defectives, but they do not adapt to different process variabilities. Pre-control is the same for processes with capability C,, equal to 1.33 or 1.67, but, ideally, these processes would be handled differently. Figures 6 and 7 explore the relationship between the probability of a signal using Classical and Two-stage Pre-control and the probability of a defect. Figure 6 is a contour plot showing the probability of a defect for various combinations of p and o,assuming USL = 1 and LSL = -1. The probability of a defect, or a nonconforming unit, is given by P(red) in Eqs. (1). Figure 7 shows contour plots of the probability of a signal for Classical and Two-stage Pre-control [unity minus the probability of acceptance as given by Eqs. (3) and (4)]. Ideally, the contours in Figures 6 and 7 would look similar; however, this is not the case. The Pre-control methods are too conservative when o is small, because they signal often for many processes that yield a very small proportion of defects. This is because, using Pre-control,

I'

0 *l a f 20

Table 2. Modified Pre-control Compared with Control Charts

MODIFIED PRE-CONTROL

X CHART

n =3

R CHART

n=4

7 CHART

n=5

Psign.~

prignal

Psignal

psi@

,0238 ,2097 ,8370

,0027 ,1024 ,6787

,0027 ,1587 ,8413

,0027 ,2225 ,9295

STEINER

Downloaded By: [Canadian Research Knowledge Network] At: 18:16 23 November 2010

-1

0.5

0

0.5

M~UU

Figure 6. Contour plot of the probability of a defect.

large p values together with small o values likely lead to many yellow units and, thus, a signal. As a result, Precontrol is not applicable when o is very small compared with the specification limits-say, when 60 covers less than 60% of the tolerance range.

Acceptance control charts do not suffer from this shortcoming because they adjust their control limits for different o values. Table 3 gives specific values shown on the contour plots and corresponding values from an ACC with n = 5 for a direct comparison. The results for ACCs are derived assuming that the estimated mean and standard deviation values are equal to the true values. Most enlightening are the two cases p = 0, a = 0.2, and C( = 0.6, o = 0.1. In both cases, the probability of a defect is fairly small and approximately the same. However, when p = 0 and o = 0.2, the probability the Pre-control scheme signals is very small, whereas when p = 0 . 6 and o = 0.1, the probability of a signal using Pre-control is very large.

PRE-CONTROL AND SIMPLE ALTERNATIVES

Table 3. Probability of Defect and Signals

CLASSICAL

TWO-STAGE

PRE-CONTROL PRE-CONTROL

P

a

P(defect)

Psi&

Psi&

0

0.1

-0

0

0.2

5.7E-5

0

0.3

8.68-4

0.5

0. I

2.9E-7

0.6

0. I

3.2E-5

0.7

0.1

,0013

3.2E-13 1SE-4

,0099 .25 .7079 ,9550

1.7E-18 1.8E-5

,0088 .4688 ,9540 ,9994

ACC n = 5

Psipa!

-0 2.6E-7

Not applicable 1.3E-7 ,0018 ,2489

Downloaded By: [Canadian Research Knowledge Network] At: 18:16 23 November 2010

In contrast, for ACCs, the probability of a signal is very small in both cases.

In summary, Classical and Two-stage Pre-control are fairly competitive with ACCs, except under certain circumstances. When the process variation is very small compared

with the tolerance range (60 < 0.6T, where T is the tol-

erance range), Classical and Two-stage Pre-control lead to undesirable results such as rejecting a process that is producing virtually all parts within specification. Also, as discussed in Ref. 3, when the process variation is relatively

large compared to the tolerance range (60 > 0.88ZJ, Clas-

sical and Two-stage Pre-control yield excessively large false alarm rates. However, if 6 0 lies between 60% and 88% of the tolerance range, Classical or Two-stage Precontrol yield good results. Modified Pre-control, on the other hand, is similar to Two-stage Pre-control with 6 0 equal to 100% of the tolerance range. As a result, Modified Pre-control is a poor method because it yields too many false alarms.

Alternatives to Traditional Pre-control

As discussed in the previous section, Pre-control has a number of important advantages over traditional control charts, mainly in terms of simplicity of implementation. However, its design choices, in particular the grouping criteria and the decision rules, appear quite ad hoc. We may ask if the performance of Pre-control could be improved while retaining its simplicity? This section considers three simple variations of Pre-control called Ten Unit Pre-control, Mean Shift Pre-control, and Simplified Precontrol. Each of these proposed variations is very similar to Two-stage Pre-control and requires only minor modifications. The goal is not to develop the optimal approach under particular assumptions but rather to consider simple changes that retain the flavor of Precontrol. Optimal ACCs and Shewhart-type charts for grouped data under distribu-

tional assumptions are developed by Steiner et al. (9.10). utilizing the likelihood ratio.

One reason why two-stage Precontrol is fairly competitive with ACCs in terms of power is due to its partially sequential decision procedure. As a result, one possible improvement approach is to make the decision procedure more sequential. A totally sequential procedure is feasible theoretically but would be difficult to implement because it would require large sample sizes occasionally and defining an appropriate acceptance/rejection criterion would be difficult. A version of Pre-control that allows a maximum of ten units at each decision point is a compromise. The decision rules for proposed Ten Unit Precontrol are given below. To monitor the process, this decision procedure should be repeated periodically.

Take one sample unit at a time.

Define #G and #Y as the cumulative number of green and

yellow units, respectively.

Stop the process if -There are any red units, or

-At any time, #Y - #G 2 2 together with #Y

2 3, or

-At any time, #Y t 5

Continue operation of the process, and stop sam-

pling, if at anytime #G - #Y 2 2.

Otherwise, continue to sample.

Expressions for the probability of a signal and the average sample size of Ten Unit Pre-control are derived by enumerating all the possible paths to acceptance and rejection. Table 4 shows a comparison of k o p e r a t i n g characteristics of the proposed Ten Unit Pre-control scheme and Two-stage Pre-control for various combinations of the process mean and standard deviation. Table 4 shows that Ten Unit Pre-control has lower false alarm rates and more power to detect large shifts in the mean than Two-stage Pre-control. In addition, although the Ten Unit Precontrol

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download